(*^
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Exercise 1: Euler rotations
;[s]
1:0,1;28,-1;
2:0,19,14,Times,1,18,0,0,0;1,19,14,Times,1,18,65535,0,0;
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = text; inactive; preserveAspect]
Let's start by creating a 3-D rotation matrix, ry[th_],  that generates rotations about the y-axis.
:[font = input; preserveAspect]
ry[th_]={{Cos[th],0,Sin[th]},{0,1,0},{-Sin[th],0,Cos[th]}};
:[font = text; inactive; preserveAspect]
Verify that this is a unitary transformation, that is: ry[th].Transpose[ry[th] ] = unit matrix
;[s]
4:0,0;55,1;81,0;94,1;95,-1;
2:2,16,12,Times,0,14,0,0,0;2,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
ry[th].Transpose[ry[th] ] //Simplify //MatrixForm
:[font = text; inactive; preserveAspect]
Similarly, let's create a 3-D rotation matrix, rz[th_],  that generates rotations about the z-axis.
:[font = input; preserveAspect]
rz[th_]={{Cos[th],Sin[th],0},{-Sin[th],Cos[th],0},{0,0,1}};
:[font = text; inactive; preserveAspect]
Verify that this is a unitary transformation, that is:rz[th].Transpose[rz[th] ] = unit matrix
;[s]
3:0,0;54,1;80,0;94,-1;
2:2,16,12,Times,0,14,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
rz[th].Transpose[rz[th] ] //Simplify
:[font = text; inactive; preserveAspect]
Eulers matrix can be define as:
:[font = input; preserveAspect]
euler=ry[th1].rz[th2].ry[th3];
:[font = text; inactive; preserveAspect]
Here is a rule that generates random angles between [0,2 Pi].
:[font = input; preserveAspect]
rule=
{th1-> 2 Pi Random[]
,th2-> 2 Pi Random[]
,th3-> 2 Pi Random[]
};
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We can use this to generate a random Euler matrix:
:[font = input; preserveAspect]
mat= euler /.rule //N;
mat //MatrixForm
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There is a theorem that state that the Euler matrix always has one real eigenvector
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ev= Eigenvectors[ mat //N ] //Chop;
ev //MatrixForm
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and at least one eigenvalue that is equal to 1. 
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Eigenvalues[ mat //N ] //Chop
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This implies that an Euler rotation can always be described as a simple rotation about a eigenvector with eigenvalue =1. 
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Problem 2: Euler rotations
;[s]
1:0,1;27,-1;
2:0,19,14,Times,1,18,0,0,0;1,19,14,Times,1,18,65535,0,0;
:[font = subsection; inactive; Cclosed; preserveAspect; cellOutline; fontColorRed = 65535; startGroup]
Your Challenge: For a given numerical Euler matrix, find the rotation axis and angle that will generate the Euler matrix.

Bonus: Can you find a similarity transformation that converts mat into a simple rotation about the z-axis. 
;[s]
2:0,0;123,1;231,-1;
2:1,16,12,Times,1,14,65535,0,0;1,16,12,Times,3,14,65535,0,0;
:[font = text; inactive; preserveAspect; fontColorRed = 65535]
I left my comments in if you want to follow my method; or you can choose your own method.
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Let's first identify the eigenvector that is invariant under the mat operation.
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Now, let's check that vec is normalized:
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Now, let's find a vector, perp, that is perpendicular to vec. 
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We do this by taking an arbitrary vector, and subtraction off the component parallel to vec. 
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Let's verify this is perpendicular:
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Let's normalize perp:
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perp2 is perp rotated by mat
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The Cosine of the angle between perp and perp2 gives us the rotation angle. 
^*)